Talk:Fermat's Last Theorem/Archive 1

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Archive 1 Archive 2

This archive of Talk:Fermat's last theorem covers the years 2002-2006.

Old or undated discussion

Why was this page moved to a lowercase title? Fermat's Last Theorem was not Fermat's last theorem - in fact, it wasn't one of his theorems at all. --Zundark 12:43 Oct 30, 2002 (UTC)

See Wikipedia talk:Naming conventions (theorems)

Is it not inconceivable that Fermat was joking about the proof fitting into the margin? Surely he proposed the equation based on a hunch he had but the proof eluded him - effectively laying down the gauntlet for others to attempt a proof.

Surely, mathematicians have a sense of humour too?

Natalinasmpf: Well, for one, Fermat's notes were in his private copy of the Arithmetica. I do not think he would be laying down the gauntlet when it was kept secret, and thus probably not intended for public display.

I agree, FLT should be all uppercase. No big deal though.

Fermat may have thought that he had the proof when he wrote the note in the margin--there are ways to "prove" the theroem that are now known to be wrong, and Fermat may have done one of them and later realized his error (which would explain why he never published his proof).

What does worked "in isolation", regarding Wiles, mean exactly? It's utterly meaningless and misleading. Wiles had plenty of human contact during that period. Not only that, he found time to advise Ph.D. students, interact with other faculty in the department, and publish several important papers!

Certainly he kept his work on FLT secret but is that a sign of isolation? Let's not let the Wikipedia degenerate into recopying exaggerated statements made in the popular media.

Since nobody has responded to my comments, I am going to adjust the entry.

Theres an excellent book on the subject, called Fermat's last theorem (funnily enough!) by Simon Singh if you want to learn the full history of the subject and Wiles eventual proof.

Yep, it's great, I've read it mroe than once. And it's listed in the article in the Bibliography section. ✏ Sverdrup 17:12, 12 Oct 2004 (UTC)

I would remove the note about Russian professor Alexander Ilyin, until his proof is really published. Anyway, his proof shown on TV doesn't look correct, so I doubt it will ever be published. 03:22, 24 August 2005 (UTC)

OK, but I would still add the fact that a short version of the theorem is still the focus of many because the way this section of the article is structured, one gets the impression that the writer(s) disbelieve that Fermat ever did have a short version. If you want, you can modify the sentence I added and take away any reference to Ilyin and merely state that many people are still looking for this shorter proof and that many people still believe it exists. 07:53, 24 August 2005 (UTC)

Did Fermat Have a Proof (2)

Proof claims not coming from within the mathematical establishment traditionally meet with some hostility. Rejection of unsolicited papers without reading is often enough used by leading mathematicians to defend their time.

What does this mean? Does it really have anything to do with FLT, and is it NPOV? -- 22:06, 21 Jan 2005 (UTC) aka Etimbo | Talk 22:07, 21 Jan 2005 (UTC) when logged in (oops)

I've noticed this as well, and was about to list it for deletion in this talk page. Seeing that other people are baffled as well, I have removed this sentence. -- Itai 13:16, 24 Jan 2005 (UTC)

Star Trek

The quote from Star Trek is wrong, afair. I don't have the exact dialogue here, but I think the writers cleverly made it vague whether or not the theorem is proven in 23xx.

I own the DVD of the episode, and have just coppied down the subtitles of their conversation for reference:
Picard: Fermat's Last Theorem. You familiar with it?
Riker: Vaguely. I spent too many math classes daydreaming about being on a starship.
Picard: When Pierre de Fermat died, they found this equation scrawled in the margin of his notes-- "x to the Nth plus y to the Nth equals z to the Nth where n is greater than two" which he said had no solution in whole numbers but he also added this phrase, "remarkable proof."
Riker: Yea, that's starting to come back to me. There was no proof included.::
Picard: Mm-hmm. And for 800 years people have been trying to solve it.
Riker: Including you.
Picard: I find it stimulating. Also, it puts things in perspective. In our arrogance, we feel we are so advanced, and yet we cannot unravel a simple knot tied by a part-time French mathematician working alone without a computer.
Riker: Captain, we've detected...
Looks to me as though it's pretty clear that they're unable to prove it. =P --The Computer Mutt 05:48, May 23, 2005 (UTC)

The Simpsons

In the "Homer in 3D" story from Treehouse of Horror IV, a four-digit disproof of FLT is just one of many mathematical easter eggs in the abstract space Homer discovers behind a bookshelf. counterexample

a. Wiles proved Fermat's Last Theorem. If his proof is correct, no disproof can be found. As one who is not an expert in the field, I accept that his proof is correct, because hordes of number theorists, group theorists and other assorted mathematical geniusses will have pored over his paper for hours; and because he had to correct an error in it. So it's been through that rigorous process and had to have changes made already.
b. If a disproof could be found that simply, do you not think people would have given up a while ago?
c. The claimed disproof is not, because it's false, as a simple check will show. The identity given (for others' reference) is
1782^12 + 1841^12 = 1922^12
Now, the last digit of 1782^12 will be the same as the last digit of 1922^12, as both end in the same digit (in fact, it is 6). But the last digit of 1841^12 is not zero (in fact, it is 1). So the left-hand side ends in a different digit from the right-hand side. Your identity is not. As for P = NP, well, the jury's still out on that one. Wooster 11:51, 19 July 2005 (UTC)

Fields medal

Actually, there are only three number theorists who supposedly went through Andrew Wiles "proof", Ribet, Mazur and Zarnak.
The main problem with Wiles approach is that he jumped into the problem without looking at the status of the underlying fields, namely, foundations, particularly, mathematical reasoning, number theory and the real number system. I have discussed those elsewhere on this page. One requirement for a consistent mathematical space is that every concept must be well-defined, i.e., its existence, properties or behavior and relationship with other concepts are specified by the axioms. In the case of a decimal every digit must be known or computable, i.e., there is some algorithm or rule that determines it uniquely. Most of the real numbers do not satisfy these requirements. There are other defects of the real numbers that I have already discussed elsewhere on this page. Consequently, the real number system is ill-defined and any problem formulated in it including FLT is also ill-defined and nonsense. Therefore, any proof of it does not make sense.
The first crucial step in the resolution of FLT is an overhaul of these underlying fields which I did the essential parts of which I have discussed elsewhere on this page. However, interested viewers may look at the original sources listed elsewhere here as well as the relevant articles on my website, . I shall be glad to answer specific questions that I have not answered yet on this page. E. E. Escultura

I have removed the following paragraph

Amazingly enough, if his original proof had been correct he most certainly would have won the most famous award in mathematics, the Fields Medal. However mathematicians must be under 40 years old to receive the award, which he had passed by the time the successful proof was submitted.

Andrew Wiles was born on 11 April 1953. He presented the original (incorrect) proof June 21-21 1993. Therefore he was over 40 years old when he presented the original proof and hence ineligible for the Fields Medal. --Etimbo | Talk 13:29, 6 Apr 2005 (UTC)

For what it's worth, the Fields medal may be awarded to any mathematician no older than 40 in the year the prize is given: for example, Perelman was 40 this year. So if the Fields were given in 1993, it would have been fine; however, he would have had to wait a year anyway, so he wouldn't have gotten it. Ryan Reich 21:06, 13 October 2006 (UTC)

Looking for false proofs.

As I understand it, there are thousands of false-proofs for the FLT. I am trying to collect some of them. They are very interesting, from an educational point of view, because you can see what are the most common deduction errors people make. In the net I haven't found much information (sites about the real proof are aboundant, and probably not many people cares about false deductions). Do you have some references?

I will appreciate an email: H. Flores ""

Try the sci.math newsgroup. Charles Matthews 16:46, 8 September 2005 (UTC)

The Last Proof?

See in the personal web ( an original evidence of Fermat's Last Theorem using only square and rectangular figures. This way we can achieve of simpler form the same conclusion to which Andrew John Wiles came up in 1995.

And in the Forum (in Spanish):

Fernando Moreno

Link to Gödel's incompleteness theorem

So, what is the connection to Gödel's incompleteness theorem? I lent out Singh's book, so I can't check it there. Our article on Gödel's incompleteness theorem does not mention Fermat, so a link does not seem very useful, even if there is a connection. The only think I can think of is speculation like "wouldn't it be cool if FLT was actually unprovable?" (before Wikes' proof). -- Jitse Niesen (talk) 23:40, 6 December 2005 (UTC)

Before Wiles' proof, FLT was often used as an example of a theorem that might be undecidable, in the sense that it might have had neither a counterexample nor a proof. Of course, now we know there is a proof for FLT - and the role of "most popular candidate for an undecidable theorem" seems to have been taken over by Goldbach's conjecture. I can't see any current connection between FLT and Gödel's incompleteness theorem. Gandalf61 09:44, 7 December 2005 (UTC)
Douglas Hofstadter connects them in his book Gödel, Escher, Bach. There, on of his characters (the Tortoise?) states that he has proven the FLT to be undecidable. I don't suppose Hofstadter actually belived FLT was really undecidable, rather he used it as an example, as you mention above. Take all this with a grain of salt though as I don't have the book and read it (the translation in fact) long ago. -- B jonas 15:01, 14 December 2005 (UTC)

The question is, is that enough to include Gödel's incompleteness theorem in the See also section? I don't think so, but apparently others differ. Ah well, it's not the most important thing on the world, or even Wikipedia. -- Jitse Niesen (talk) 17:41, 14 December 2005 (UTC)

Actually, none of the comments above seems to push for inclusion. And I agree with too, and would just remove it. It can only confuse a reader who will think that there is some link between the two topics. Schutz 18:01, 14 December 2005 (UTC)
There is a link, see what is now the bottom of this page. 10:26, 29 March 2006 (UTC)

The best way to have informed discussion on the subject is to look at the original documents. Cited below are Wiles' and Taylor's work on Fermat as well as Escultura's partial list of his papers on the critique-rectification of the real number system that paved the way for the construction of countable counterexamples to FLT and applications.

[129] Wiles, A. (1995) Modular Elliptic curves and Fermat's last theorem, Annals of Mathematics, 141(3), 443 – 551. [130] Wiles, A. and Taylor, R. (1995) Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141(3), 553 – 572.

[18] Escultura, E. E. (1996) Probabilistic mathematics and applications to dynamic systems including Fermat's last theorem, Proc. 2nd International Conference on Dynamic Systems and Applications, Dynamic Publishers, Inc., Atlanta, 147 – 152. [20] Escultura, E. E. (1998) FTG VII. Exact solutions of Fermat's equation (Definitive resolution of Fermat's last theorem, J. Nonlinear Studies, 5(2), 227 – 254. [29] Escultura, E. E. (2002) FTG V. The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169. [31] Escultura, E. E. (2003) FTG XVII: The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149. [35] Escultura, E. E. FTG XV. The new nonstandard analysis and the intuitive calculus, accepted, International Journal of Nonlinear Differential Equations. [36] Escultura, E. E. (2002) FTG VI. The philosophical and mathematical foundations of FLT’s resolution, rectification and extension of underlying fields and applications, accepted, J. Nonlinear Differential Equations. [37] Escultura, E. E. (2002) FTG XXII. Extending the reach of computation, accepted, International Journal of Nonlinear Differential Equations [42] Escultura, E. E. (2004) FRG XXVII – XXVIII. Part I. The new frontiers of mathematics and physics. Part I. Theoretical Construction and Resolution of Issues, Problems and Unanswered Questions. [43] Escultura, E. E. (2005) FRG XXVII – XXVIII. The new frontiers of mathematics and physics. Part II. The new real number system: Introduction to the new nonstandard analysis, Nonlinear Analysis and Phenomena, II(1), January, 15 – 30. [45] Escultura, E. E. FTG. XXVI (2005). The new nonstandardd calculus, accepted, Nonlinear Studies. [47] Escultura, E. E. (2006) FTG XXXIV. Foundations of Analysis and the New arithmetic,Nonlinear Analysis and Phenomena, January 2006. [48] The Pillars of the new physics, Nonlinear Analysis and Phenomena, Vol. III, No. 2.

Nexus numbers paragraph

I removed the following paragraph (contributed by an anon) from the main article because it is unsupported and difficult to follow. If anyone can find a reference and re-write it in a more intelligible manner, it could be re-inserted ... Gandalf61 14:28, 24 February 2006 (UTC)

Probably Fermat studied the powers of integers as sums of adjacent nexus numbers. If a table of power of n is written in form 1+ (2^n-1^n) + (3^n-2^n) + etc. + [X^n - (X-1)^n], i.e. 1+ sum of adjacent nexus numbers, it can be noticed that a difference of the same power of two integers is always a nexus number or a sum of adjacent nexus numbers. Also it can be noticed, that only in the table of squares (n=2) there exist squares of integers among the nexus numbers and sums of adjacent nexus numbers. Opposedly, in the table of cubes (n=3) there are never cubes of integers among the nexus numbers and sums of adjacent nexus numbers. Accordingly, in the power tables higher than cubes (n>3) there are never powers of integers among the the nexus numbers and sums of adjacent nexus numbers, since the powers higher than cubes are actually lines of cubes. Thus a difference or a sum of the same power of any two integers can not be the same power of third integer, if n>2.

Apparently, any references can not be found, but FLT seems self-evident if power sequences are compiled as sums of nexus numbers: Reading of powers always start with 1+ etc. while the differences always consist of only a right end fraction of the nexus sum representing the power of n of the larger integer. Can you formulate it more intelligible manner? The nexus line may really be the "demonstrationem mirabilem" which was too large to be presented on the margin of Diophantus. Demonstration may be presented as shown below:

Powers of n |------------->----------->------------------>------------------------------------->

X^n  =        1 + (2^n - 1^n) + (3^n - 2^n)+ etc.+ [Y^n - (Y-1)^n] + [(Y+1)^n - Y^n]  +etc. + [X^n - (X-1)^n]
                         |<-------<----------<-------------------<-------------------------------------|  Differences of powers 08:53, 26 February 2006 (UTC)


Shouldn't this be renamed "Wiles's theorem"? —Keenan Pepper 16:31, 24 March 2006 (UTC)

The theorem? Sure, if you ask me. But not the article. FLT is the most common name by far. Fredrik Johansson 16:47, 24 March 2006 (UTC)

Not provable in ZFC?

He uses topoi. You can't prove they exist. It's equivalent to assuming the existence of an inaccessible cardinal. Incidentally this partially address what that rambler wrote at the beginning of this top page; it very well may be unprovable in ZFC. If you need that it is consistent that an inaccessible cardinal exists in order to prove FLT then by Goedel you can't prove it from ZFC, period. In fact the possibility that it is unprovable in ZFC is an extremely interesting one to logicians, however the consensus seems to be that it should be possible to compress Wiles' proof to third order number theory (much much weaker than ZFC, stronger than what's used in analysis). I personally haven't read the proof but Solovay is the original source of what I said, so we can assume that it is true. 10:24, 29 March 2006 (UTC)

Algebraic geometers usually formulate schemes in terms of categories, and categories in terms of NBG set theory. Since NBG is a conservative extension of ZFC, with all of the theorems about sets the same, this is not much of a leap.
The above assumes, without argument or much reason, that if you can't prove it in ZFC then it's somehow dubious, but using NGB or assuming a Grothendieck universe and hence an inaccessible cardinal is a very mild extension of ZFC, which is not a totem object to be worshiped as some sort of god. In any case, it's hardly likely that the full power of NGB is required, or the full power of ZFC for that matter. It would be an interesting project to try to reformulate enough of algebraic geometry in first order arthmetic to get it to work, as I imagine could be done; I doubt you would need second order arithmetic to prove it. Maybe Woodin (since Solovay is retired) and Ribet should get together and think about it, but I doubt that will happen either, since it would be a huge mess worthy of someone's thesis.

Gene Ward Smith 22:23, 3 May 2006 (UTC)

Oh, it would be interesting to get it to work in first order arithmetic, would it? Oh, you imagine it could be done, do you? See what Solovay said about it at the FOM list. I believe it was March or April. He says that Wiles proof could not be compressed below third order number theory as it stands. No doubt you know better and could prove the entire thing in first order arithmetic if you tried, though Fermat, Gauss, Euler etc. couldn't. 01:40, 10 May 2006 (UTC)
The problem with ZFC is the inclusion of the axiom of choice involving the universal and existential quantifier on infinite set which is unverifiable and does not endow certainty to a theorem. E. E. Escultura
The existence of ANY infinite set is unverifiable, as is the existence of very large finite natural numbers.

Observation by a non-mathematician

Mr. Escultura's "disproof" (I think it is) upstairs is rather confusing for me, but I noticed 1 glaring mistake: his "dark number" is actually 0. I'm familiar with the proof that 0.999... = 1 and as "a published mathematician" he should know it too. Let x=0.999... then 10x= 9.999... and 10x - x = 9x = 9.000... so x=1 And this guy has been published in journals!? He sounds like a circle squarer... Sorry if I misread the text (very likely!) Zyxoas 11:07, 29 March 2006 (UTC)

Mm-hm, you're quite right in all particulars, except the mis-reading part. Just read Andy Wiles' e-mail to the guy; it's a bit of a hoot. Anyone who claims that "most of the concepts of mathematics today are ill-defined." deserves what's coming to him; I reckon that's good for about fifty points on the fabled Baez Crank-o-meter, going by the number of mathematical physics topics he thinks ought to be blown out of the water. Wooster 15:50, 31 March 2006 (UTC)
First, some preliminaries. The concepts of individual thought cannot be the subject matter or objects of a mathematical space because they are not accessible to others and can neither be studied collectively nore axiomatized. Therefore, the objects of a mathematical space should be symbols (we call them concepts, too) well-defined by consistent axioms. A concept is well-defined if its existence, properties or behavior and relation with other concepts are specified by the axioms. Existence is necessary because vacuous statement or defining expression of a concept is inherently contradictory. For example, the concept triquadrilaterial, i.e., a plane figure with three edges and four vertices, is vacuous, therefore, nonsense. In the construction of the new real number system R*, I take the symbols as decimals and call them new real numbers) constructed on the basic integers 0, 1, ..., 9 which I take as elements of R* (insures their existence) the other requirement being met by the addition and multiplication tables of elementary arithmetic. The rest of the new real numbers are constructed as decimals, where we first define 10 = 1 + 9 and, say, 239.578 = 2(10^2) + 3(10) + 9 + 5/10 + 7/10^2 + 8/10^3, etc. Subtraction and division are the inverse operations to + and x, respectively. Then the new real numbers are well-defined and have natural ordering (the real numbers have none). Since 1 and 0.99... are distinct symbols and are, therefore, distinct decimals and in their natural ordering 1 > 0.99..., the number d* = 1 - 0.99..., called dark number, a nonstandard number, is positive. Therefore, the statement:
"Let x=0.999... then 10x= 9.999... and 10x - x = 9x = 9.000... so x=1", is nonsense. Obviously, we cannot rely on ::high school mathematics and, BTW, name-calling cannot resolve one's ignorance of the subject.
Since a number and its properties are determined by its digits there is additional requirement for a new real number to be well-defined: every digit is known or computable, i.e., there is some rule or algoritm for determining it uniquely. The irrationals pi and the logaritmic base e are well-defined because their digits can be computed from their series expansion. The decimal 3.1..., where every decimal digit is the last digit of the sum of the preceding decimal digits is also well-defined (whether this is a rational number is an open problem). So is a normal number where each digit is taken at random from the basic integers.
We cannot add or multiply nonterminating decimals (we can only approximate the sum or product) because either operation requires the last digit to carry out. This tells us that nonterminating decimals are ill-defined in the real number system. That is why the calculator or computer cuts initial segments in computation. If we insist that they are then we have this contradiction. Recall this elementary theorem in arithmetic: The average between two unequal numbers lies strictly between them, i.e., if a > b then > (a+b)/2 > b. Let a =1, b = 0.99..., Then

(1.99... )/2 = 0.99... = b, contradiction. E. E. Escultura

I stopped reading when you said ``We construct R* as follows, constructing its elements using the digits 0 through 9. We define these numbers to be elements of R*, which ensures that they exist." 19:40, 29 September 2006 (UTC)

About the incomplete totality of the set of all prime natural numbers

Essay moved to User:BenCawaling/Essay. Gandalf61 08:26, 14 April 2006 (UTC)

So, is this another circle-squarer? Admittedly I didn't read the entire novela but I became suspicious when I saw something like: "let N be an infinite set and let P(N) be its power set; let F:N<-->P(N) be a 1-1 mapping between N and P(N)..." how can there exist a 1-1 mapping between a non-trivial set (order > 1) and its power set? He also seems to confuse infinite cardinals (Aleph-NULL, the Continuum etc) with the infinite ordinal (Infinity), or am I talking absolute kakas? Zyxoas (talk to me - I'll listen) 17:21, 15 April 2006 (UTC)

It is clear that you are not a learned mathematician nor are you familiar with "proof by contradiction" (in fact, I was merely restating Cantor's theorem's "proof"). It would certainly help if you read and understand what you are commenting on ... []BenCawaling 09:09, 2 June 2006 (UTC)

The first Simpsons counter-example

The article claims that the LHS and RHS are different "within a billionth of each other" - that doesn't sound right. A billion is at most 12 (13) digits, and the numbers differ by more than that. Zyxoas (talk to me - I'll listen) 19:46, 25 April 2006 (UTC)

The relative difference 1-1/(x-y) is 3 × 10-10, or one-third part in a billion (109). Count digits from the top. Fredrik Johansson 20:20, 25 April 2006 (UTC)

Wiles' Proof Wrong?

A news article was published recently which seems to indicate that the 1994 proof from Princeton is incorrect. This should be studied and ammended by someone as appropriate. --Sr.Wombat 5/11/2005

No. Read letter by Dr. Wiles carefully: he is being sarcastic. The guy from Manilia is evidently a known crank. In any event the news article defines integer incorrectly which might be a hint. Also notice that Escultura's supposed disproof uses a different number system. Fermat's last theorem is for the typically used number system and thus any proof or disproof in it is simply irrelevant.

No, No. A mathematical issue cannot be resolved by anyone's letter. The Manila Times article says, UP Prof Proves Princeton Man Wrong, and proceeded to list the papers in renowned refereed international journals (not hometown journals) that accomplished this feat. E. E. Escultura
Those who bring in Wiles in the discussion is ignorant of the issue. The Manila Times article says, UP Prof Proves Princeton Man Wrong, and proceeded to enumerate the papers in renowned refereed international journals where this feat is accomplished. The guy who brings in Wiles' letter takes his ideas not from the top but from the flat of his foot. For those interested in the subject find leads to original source on my website: (E. E. Escultura).

The problem with all that is that the mathematics community does not agree with E. E. Escultura and all he has done is say that any mathematics that is not based on his axioms are incorrect. Please see

The final arbiter of scientific knowledge in the mathematics community is the network of refereed international scientific journals which published over two-dozen papers on my work. Internet messages don't count at all. Refutation of anything published should be published in similarly refereed journal. Failure to do so is acceptance by default. The Manila Times article is quite clear, UP Prof Proves Princeton Man Wrong, and it proceeded to mention the scientific journals where this feat was accomplished. The issue here is simple: the status of FLT and Wiles' "proof". The answer is quite categorical: FLT is false and Wiles is wrong. The problem with Alex Pabico of PCIJ is: he is totally ignorant of the issue and yet he roams around the internet commenting on something he knows nothing about and confuses the science community. That is why that blog is in trouble with the law; it is quite irresponsible. E. E. Escultura

One of the most important things to note is that FLT was targeted for its popularly(sp?) and that if Wiles were wrong then all the mathematics that he based his work on and the mathematics based on his would not be true. The real number system is taught to all research mathematicians and is what is respected. Timothy Clemans

Fundamental Defects in the Theory of Mathematics?

For the benefit of the experts I would like to share my strategy for capturing FLT.

1) I wondered why this conjecture could not be resolved for centuries and I concluded that present mathematics is defective or, at least, inadequate. So I embarked on a critique of its underlying fields, namely, foundations, number theory and the real number system.

2) In foundations here is what I found:

a) I agreed with David Hilbert that the concepts of individual thought cannot be the subject matter of a mathematical space because they are not accessible to others and can neither be studied collectively nor axiomatized. His remedy was to represent them by symbols well-defined by a consistent set of axioms. A symbol or concept is well-defined if its EXISTENCE, properties and relationship with other symbols are specified by the axioms. I stress existence because vacuous concepts are contradictory as the following definition shows: A triquadrilateral is a plane figure with three vertices and four edges.

b) Other ambiguous or uncertain concepts or propositions, which are sources of contradiction, include: large or small number (depending on context), ill-defined concepts, infinite set and self-reference. An example of self-reference is Richard paradox: The barber of Seville shaves those and only those who do not shave themselves; who shaves the barber? Incidentally, the indirect proof is self-referent. The remedy is constructive proof. We may introduce some ambiguity in a mathematical system provided it can be approximated by certainty. For example a nonterminating nonperiodic decimal is inherently ambiguous but we can approximate it by a segment up to the nth digit and the margin of error will be 10^-n.

c) Any proposition about ambiguous set (e.g., infinite set) involving the universal or existential proposition is not verifiable and cannot be an axiom of a mathematical space since it would not endow certainty to the conclusion of an axiom.

d) The following items are required for a contradiction-free mathematical space:

i) It must be well-defined by consistent set of axioms, meaning, every concept must be well-defined. ii) A decimal is well-defined if every digit is known or computable, i.e., there is some algorithm or rule for determining it uniquely. iii) The rules of inference must be specific to and well-defined by its axioms.

3) The real number system does not satisfy the requirements of a contradiction-free mathematical space and is, therefore, nonsense. Consequently, FLT presently formulated is nonsense. Here are some of the defects of the real number system:

a) Most of its concepts are ill-defined. b) The trichotomy axiom that says, given two real numbers x, y, one and only one of the following holds: x < y, x = y, x >y, is false. A counterexample to it is a pair of normal numbers. A normal number is constructed by taking every digit from the basic integers, 0, 1, …, 9, at random. Brouwer also constructed a counterexample to it that proved, at the same time, that the real numbers have no natural ordering. c) The completeness axiom is unverifiable because it involves the universal and existential quantifiers infinite real numbers.

4) The remedy for the real number system is to reconstruct it as decimal numbers R* subject to the following simple axioms R*:

a) R* contains the basic integers. b) The addition table. c) The multiplication table

Then FLT, reformulated in R*, has countable counterexample posted elsewhere on this website.

5) With respect to number theory its main flaw is that the integers, its subject matter, have no valid axiomatization. The remedy is to embed them into R* isomorphically as the integral parts of the decimals.

The counterexamples to FLT in the contradiction-free new real number system are as follows:

Let x = (0.99…)^T, where T = 1, 2, …, is an ordinary integer. y = d* = 1 - 0.99…, called dark number. z = 10^T. Then, for n > 2,

x^n + y^n = z^n. E. E. Escultura

The new real numbers x, y, z (the well-defined real numbers form a subspace of the new real numbers) are counterexamples to FLT. Moreover, the the countably infinite triples (kx,ky,kz), k = 1, 2, …, are also counterexamples to FLT since

(kx)^n + (ky)^n = (kz)^n.

Together, they constitute the countably infinite counterexamples to FLT.

These counterexamples are constructed from the new integers, namely, d* and numbers of the form N.99…, where N = 0, 1, ... is an ordinary integer, and the ordinary integer of the form 10^T. They form a subspace of the contradiction-free new real number system. The mapping, 0 -> d*,

N -> (N-1).99…, N = 1, 2, …,

is an isomorphic mapping of the ordinary integers into the new integers. This means that they have almost the same property the only difference being that d* > 0. The mapping of the well-defined real numbers, i.e., the terminating decimals, into the periodic decimals having all 0's beyond a certain digit embeds the integers isomorphically into the new real numbers and subject them to the axioms of the latter. This resolves the fundamental flaw of number theory that the integers, its subject matter, are ill-defined. In other words, until this embedding, the integers were nonsense. For details see my website, . E. E. Escultura

-- Unsigned comment. Added on 16 March 2006 by IP

Summary and more foundational updates on the requirements for a contradiction-free mathematical space: 1) The subject matter of a mathematical space consists of symbols (which we also call concepts) well-defined by or subject to consistent axioms. 2) Every concept must be well-defined, certain or unambiguous; a concept is well-defined if its existence, properties or behaviour and relationship to other concepts are specified by the axioms. 3) The sources of ambiguity, uncertainty or contradiction include: a) Vacuous concept, i.e., the defining expression is vacuous, b) Large and small numbers (depending on context); this is mainly due to limitation of computation, it follows that there is only one well-defined infinity, countably infinite; the continuum does not exist. c) infinite set; the more precise concept is finite but unbounded, meaning, it cannot be contained in a finite set, d) Self-reference, e.g., Richard paradox and the indirect proof. e) Proposition about the elements of an infinite set involving the universal or existential quantifier; an example of unverifiable propostion is the incompleteness axiom of the real number system; such proposition is not admissible as an axiom because it does not provide certainty to a theorem; a false statement is inadmissible as an axiom for it introduces inconsistency to a mathematical space. 4) Since distinct mathematical spaces are well-defined only by their respective axioms, distinct mathematical spaces are independent. Therefore, a) A concept in one is ill-defined in the other, b) A proposition involving concepts from two distinct mathematical spaces is ambiguous; in particular, c) Goedel’s incompleteness theorems are flawed because they involve concepts from two distinct mathematical spaces, the propositional calculus and the integers. 5) The rules of inference must be specific to and well-defined by the axioms; it follows that universal rules of inference such as formal logic is flawed. 6) Ambiguity may be introduced in a mathematical space without creating a contradiction provided it is “approximable” by certainty; for example, although there is uncertainty in a nonterminating decimal since not all its digits are known are known or computable (this is a situation where “every” is not necessarily equivalent to “all”) it can be approximated by its segment at the nth digit and the margin of error is 10^-n. E. E. Escultura

I excerpted the following text from your “Countable Counterexamples to FLT” article in the Internet:
A common defective methodology in mathematics is illustrated by the following example:
"Theorem." The largest integer is 1.
"Proof." Let N be the largest integer; to show that N = 1. One and only one of the following must hold: N = 1, N < 1 and N > 1. By our assumption, we cannot have N < 1. If N > 1 then N*N > N and the integer N would not be the largest. Therefore, N = 1 which is no less absurd.
This is an example of the Perron paradox or the use of necessary condition without existence theory; in this case, the existence of the integer N is not established (in the rectified foundations this simply means that the concept “greatest integer” is ill-defined). Perron paradox is a variant of vacuous statement where one may state a true nonsense about anything that does not exist including its own negation.
It is clear that your placement of double quotation marks in “Theorem” and “Proof” that you do not actually believe in the above argument. Indeed, your example – both the “theorem” and the “proof” --- is plain stupid. What alarms me is your claim that this is a “common defective methodology in mathematics”. I must emphasize that your example is dim-witted and defective that is why nobody cites it (except you) so it cannot be a “common methodology in mathematics”. Indeed, what the argument establishes is “the largest integer (if it exists) is not an integer” since you can apply the same argument not only to 1 but to all the (positive) integers --- 2, 3, 4, …, ad infinitum. Perhaps, you need to familiarize yourself with the standard argument proving that 1/0 is not an element of the field of real numbers --- because if 1/0 = c is a real number then 1 = 0 * c = 0 (similarly for any non-zero real number other than 1).
Moreover, the law of trichotomy for the real numbers (or integers) applies only to _elements_ of the real number field (or integers) so you could not invoke it for “the largest integer”. The argument you presented is an instance of “modus moron” reasoning (P --> Q, ~P yields ~Q) --- hence, truly defective. What extremely appalls me is that somehow you managed to “successfully” had these nonsense published --- particularly, your ridiculous claim (following your defective argument above) that the law of trichotomy for the real numbers is flawed.
Likewise, your “dark number” and knowledge of place value positional numeral system is wanting (unacceptable even for high school students). However, your claim “Any proposition about ambiguous set (e.g., infinite set) involving the universal or existential proposition is not verifiable” as it pertains to indeterminate infinite sets (but not to determinate infinite sets which have defining functions, rules or patterns for their enumeration so that, say, mathematical induction holds for the proposition) is correct. []BenCawaling 02:24, 8 April 2006 (UTC)

The placement of the double quotation means that the "Theorem" is not realy a theorem because it is false. It is false because of the use of necessary condition without existence. This is the original statement of what is called the Perron paradox, i.e., the use of necessary condition without existence. It is a variation of the inherent contradiction in a vacuous statement. The conclusion that the greatest integer is 1 follows from the trichotomy axiom on the assumption that the greatest integer exists which doesn't. This is a very common error in mathematics: Let x be the solution ...; then x is substituted in some equation it is expected to satisfy (a necessary condition) ... You missed the main point here.

The dark number d* is a well-defined nonstandard new real number and has nonstandard Cauchy representation 0.1, 0.01, ... An infinite set has a certain level of uncertainty because of its inexhaustibility. For example if one wants to verify that every element of an infinite set A has property p, that is unachievable; in other words, the statement is unverifiable. 1/0 is simply ill-defined, nonsense. BTW, we cannot rely on high school mathematics and name-calling cannot resolve one's ignorance of the subject. E. E. Escultura

Well- The truth is that Wiles is Wrong, but he was chosen to be the person to end the Dilemma of Fermat's Conjecture.
" It is extremely amazing that it required more than 300 years after 'Perrie de Fermat'

composed, before his death in 1665, a riddle involving an elementary algebraic equation, which eluded everyone, including the greatest mathematicians, to find, in 1979, a solution. A joke? Perhaps. But, Fermat was the first to claim, while writing this riddle, that he knew the simple solution. And clearly, if this were true, which I believe that it is, then perhaps, "Fermat's Last Theorem" should rightfully be called; the greatest joke of all times. However, while I accept Fermat's claim, I do not believe that he actually knew, or fully understood, the profound implications of his discovery. Especially since, it may be concluded, as presented below, there are only 3 loghically vaible 'Interconnected Complimentary Solutions' that would solve the riddle regarding why;

                  "There are No solutions in Whole Numbers to the Equation,
                   X^N + Y^N = Z^N, when N > 2". 

               1. There is no Common Coefficient between the Sum of Two Exponents,
                  the Exponent equaling their Result, and their respective Roots,
                  when 'N > 2' , and 'N' defines the Exponent of the base variables.
                  (Equal Number of Parts Contained in the Whole.)

               2. Fermat's Solution defines how he interpretated the problem, which
                  is based upon the current mathematical knowledge known duing his
                  time: Pythagoras Theorem, and the (Analytical Geometric) solution
                  explaining the Difference regarding 'Why', when 'N = 2'; 'The Sum
                  of the Area of two Perfect Squares Equals the Area of another
                  Perfect Square'. And, when 'N = 3The Sum of the Area of two
                  Perfect Cubes is not Equal to the Area of another Perfect Cube'.
                  Nevertheless, this assumption builds an explanation that explains
                  this difference, which it is believed to be the foundation for the
                  proof that Fermat claimed would not fit in the margin of his paper,
                  but would explain why, when 'N > 2', his theorem is true.  

               3. In Exponental Operations, there is No equal Distribution of
                  Multiplication over Addition when 'N > 2', and 'N' defines the
                  value of the Exponent. (The Discovery of the Distributive Law for
                  Exponental Functions, and the Foundation for the Finite
                  Mathematical Field:
                  "The Rudiments of Finite Algebra; The Results of Quantification".)"

   {Read: ""

          Eugene Terrell
          (ETT-R&D Publications)
          Engineering Theoretical Technologies
          R&D Publications68.122.119.133 18:06, 1 June 2006 (UTC)


What? No mention or discussion of Andrew Beal and Beal's Conjecture? Pariah23 21:44, 11 February 2006 (UTC)

Actually, the counterexamples to FLT are also counterexamples to Beal's Conjecture. E. E. Escultura

I think a bigger omission is that it ought to mention somewhere in the first paragraph that Wiles has proven it! If I don't see an objection I will add that. Gene Ward Smith 22:26, 3 May 2006 (UTC)

Wiles tried to prove it but his "proof" is wrong. E. E. Escultura

Minor formatting flaw

Just as a minor note, the text in the final introductory paragraph was running under the picture in both mozilla and IE (moreso in IE). I'm a newb so I just inserted a line break between "prove any other" and "theorem), it", feel free to fix. 15:18, 8 April 2006 (UTC)

[I deleted the entire discussion on the incorrectly stated Fermat's equation due to font conversion error. E. E. Escultura]

Underlying fields

One of the major results in the resolution of FLT is an off-shoot of the critique-rectification of its underlying fields, namely, the characterization of undecidable propostion. Since two distinct mathematical spaces are independent and have noting to do with each other, a concept in one is ill-defined or nonsense in the other. Therefore, any propostion containing concepts from two distinct mathematical spaces is ill-defined, nonsense and ambiguous. One cannot prove an ambiguous proposition, it is undecidable. Naturally, Goedel's proofs of his incompleteness theorems is, at best, flawed because it involves concepts from two distinct mathematical spaces, the propositional calculus and the integers. However, the question of truth is apart from the question of decidability although what is false cannot be decidable. In the case of FLT it is both false and undecidable. E. E. Escultura
Please see Timothy Clemans 18:54, 14 June 2006 (UTC)
Compare with

"... if Wiles were wrong then all the mathematics that he based his work on and the mathematics based on his would not be true. The real number system is taught to all research mathematicians and is what is respected."

Science advances, so does its language, mathematics. What was correct before may be wrong now. Therefore, there is need for continuing reforms in math-science education. This is urgent because the wrong mathematics continues to be taught world-wide. E. E. Escultura
It is not so much that your ideas are good or bad, but that you say Andrew Wiles was wrong for not using your axioms. You only proved Andrew wrong if you prove everyone else wrong too.
I am saying that existing mathematics is defective and that FLT is nonsense and cannot be resolved in it. So I introduced the remedy that overhauled analysis, essentially, and provided the counterexamples to FLT.

E. E. Escultura

You are a fraud, because you created a hoax and still promote the idea that you proved Andrew Wiles wrong.

Name-calling cannot resolve your ignorance of new developments in mathematics. E. E. Escultura

Axioms are not true or false.

This is total ingnorance of the fundamentals of mathematics. A false axiom introduces inconsistency in a mathematical space. A century ago Hilbert spent a great deal of his life gappling with the question of consistency of mathematics and tried to prove the consistency of arithmetic which he was unable to because it isn't. Introducing a false statement as an axiom would create inconsistency and send the mathematical system down the drain. What will happen if I introduce this statement as an axiom for the plane: There exists a triangle having four edges? Contradictions will grow like mushrooms all over your mathematical space. E. E. Escultura

You do not have a good education in mathematics. You did not have any students that became professors whereas Wiles does. [1] ,

Wanna bet your life on your last statement? (One President of the University of the Philippines, several members of the board of regents, several deans and many professors).

[2]. By the way you would not have been able to find the errors in the first paper written by Wiles that the German journal found. You simply do not have the background. Timothy Clemans 00:35, 15 June 2006 (UTC)

I just pointed out to you what the fundamental errors in Wiles' work are. E. E. Escultura
BTW FLT is about non-zero integers. Somehow you do not see that. Does your real number system make it easier to prove or disprove the Birch and Swinnerton-Dyer conjecture? I did not think so. See
Your non-zero integers are ill-defined, nonsense. E. E. Escultura

Prove Ken Ribet wrong first then prove Andrew Wiles wrong second.

That is exactly what I did. E. E. Escultura

Oh, sorry but you can not do so. There are no problems with their published work on the matter. Do you even know what a representation is? Do you know about counting modular rational Galois(sp) representations and rational Galois(sp) representations? If you ever meet Andrew Wiles, I hope he is as kind to you as everyone else so you would not feel special other than the fact that meet him, but many people near the university have. Timothy Clemans 00:46, 15 June 2006 (UTC)

I find this one to be funny and yet it is important(
Ken Ribet's work is wrong because, like Wiles, his work is based on the real number system which is ill-defined, nonsense. Name-calling, name-dropping and topic dropping do not impress me. What have they to do with the status of FLT and Wiles' "proof"? When the top is empty one pulls ideas from the flat of his foot. E. E. Escultura

"According to an article published May 4th in the The Manila Times, Wiles’ proof is flawed. Well, that article could be true if we were ready to change the real number system. That’s exactly what crackpot Edgar Escultura is suggesting and provides the basis for his so-called counterexamples to Wiles’ proof. Sure Edgar, let’s reinvent the real number system and make other absurd assertions concerning the foundations on which the entire field of mathematics has evolved. Where’d you get your PhD bro? Out of a Cracker Jack box? Or are you really that good? Well, I ain’t no math genius, but I can tell you this, you’re full of shit." I do need to note that Wiles' proof of Fermat's Last Theorem took eight years of professional work using mathematics that Fermat simply could not have dreamed of. It is 20th century mathematics that one of the most diffcult mathematics works ever. Timothy Clemans 01:07, 15 June 2006 (UTC)

Mathematics is such a matured field that it can accommodate radical changes regardles of emotional outbursts and name-calling. Indeed, I have introduced radical changes to it. You need to study the original sources (references are listed here) to keep up with this new development. They are published in renowned refereed international journals (the arbiter of scientific knowledge). To shoot down my work and my criticisms of Wiles' work someone needs to publish a refutation in similarly renowned refereed journals (not hometown journals). Wiles and his three supporters, Ribet, Mazur and Zarnak(?) are quiet. My work stands by default. E. E. Escultura
Actually, I know Barry Mazur, and I suspect his silence on the matter is no such admission. He's just busy, as always, with other things :-) Dmharvey 22:04, 15 June 2006 (UTC)

Any result published in refereed international scientific journal stands as scientific knowledge unless it is refuted and the refutation is also published in similarly refereed journal. That is the way it is in science. E. E. Escultura

Summary of the properties of the new real number system:

1) It is finite (therefore, discrete) but unbounded, consistent, has natural ordering (the real number system has none), non-Archimedean (but the complement of d* and the unbounded number u* is Archimedean) and enriched by the nonstandard numbers d*, the new integers N.99..., where N = 0, 1, 2, ..., u* and the well-defined nonterminating decimals.

2) It follows that functions over the new real numbers are discrete and discrete-valued (the paper, The new nonstandard calculus, is appearing in Nonlinear Studies soon).

The new real number system is the appropriate number system for computing, especially, for simulation. It is also appropriate for physics because nature is discrete since it consists of its basic constituent, the superstring. E. E. Escultura

I like the old real number system better. Dmharvey 23:13, 18 June 2006 (UTC)

Mathematics has nothing to do with taste. E. E. Escultura

Ha ha well at least now I know why I'll always be such a hopeless mathematician. Dmharvey 21:25, 25 June 2006 (UTC)

introduction is too long

The current introduction (after edits by User:Timothy Clemans) is way too long. It needs to be refactored somehow. Dmharvey 00:14, 25 June 2006 (UTC)

proposed introduction by Timothy

The following is an introduction that I propose:

Pierre de Fermat
Problem II.8 in the Arithmetica of Diophantus, annotated with Fermat's comment which became Fermat's Last Theorem (edition of 1670).

Fermat's Last Theorem (sometimes abbreviated as FLT) is a statement about the solutions of infinitely many Diophantine equations and is one of the most popular theorems in the history of mathematics. The 17th-century mathematician Pierre de Fermat in 1637, wrote in a margin of his translated copy of Diophantus' Arithmetica next to a problem about spliting a given square into two other squares the following:

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. (It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.)

Fermat's proof of this theorem has never been found nor has his copy of Diophantus' Arithmetica been found, however his notes were published in a book by his son. Many people have tried to rediscover Fermat's proof and others have tried to simply prove it. Fermat's Last Theorem is significant because all the other theorems proposed by Fermat were settled, either by proofs he supplied, by rigorous proofs found afterwards, or by counterexamples showing a proposed theorem to be false. The theorem was not the last that Fermat conjectured, but the last to be proved. The theorem is generally thought to be the mathematical result that has provoked the largest number of incorrect proofs, perhaps because it is easy to understand.

The following is the theorem in modern language:

If n is greater than two, then an + bn = cn has no solutions in non-zero integers a, b, and c.

The only known proof of Fermat's Last Theorem was first suggested by Andrew Wiles of Princeton University in a series of lectures in 1993 in Cambridge, England following seven years of study by Wiles starting in the late summer of 1986 when a friend told him that Ken Ribet had proved a conjecture that connected Fermat's Last Theorem to a conjecture proposed by Goro Shimura based on the ideas of Yutaka Taniyama and popularized in the west by André Weil. That conjecture says that every elliptic curve over rational field was modular.

I want to be able to introduce one to FLT and Andrew Wiles and then in the bosy talk more techinical(sp). This is maybe a better introduction, but I am sure that it will need fixing up. (incorrect syntax) Timothy Clemans 21:55, 25 June 2006 (UTC)

It looks messy. Any idea would be helpful. Timothy Clemans 21:59, 25 June 2006 (UTC)

The proposed introduction does not reflect the content, essence and tone of the discussion. E. E. Escultura
It seems to me that the revision before Timothy's edits satisfies his desire "to introduce one to FLT and Andrew Wiles". Sorry to sound so critical, but I'd like to know why he was not satisfied with that version. -- Jitse Niesen (talk) 00:29, 26 June 2006 (UTC)

If this question is addressed to me, my answer is: The present formulation of FLT on the integers is nonsense because the integers are ill-defined. Even if we embed them in the real number system the problem remains because the real number system is also ill-defined. My remedy is to construct the contradiction-free new real number system and re-formulate FLT there. The new topic below enumerates the requirements for contradiction-free mathematical space. E. E. Escultura

My question was addressed to Timothy Clemans. -- Jitse Niesen (talk) 07:58, 27 June 2006 (UTC)
Dear Dr Escultura, could you please indent your comments. Please examine how we all do it here. It makes it much easier to follow a conversation. Thanks. Also you might consider getting an account. Dmharvey 10:54, 27 June 2006 (UTC)

The suggestion is well-taken. Thanks. E. E. Escultura

I was talking about the article not Fermat's Last Theorem itself. I would like to see the introduction giving more mathematical detail into FLT and the effect it has had on number theory such as the discovery of algebraic number theory.
I am concerned that this is not very good mathematics article, because in the following quote from it, the epsilon theorem is said to be stating that an elliptic curve connected to Fermat's Last Theorem exists if FLT is in-correct that would have rational coefficients, but is not modular, whereas the Taniyama-Shimura theorem saids that every rational elliptic curve is modular. I thought the whole elliptic curve existing, because of counterexamples was Frey's idea coming from the work of Hellegouarch not Serre's. I thought Serre just conjectured the exact connection between that elliptic curve and modular forms. Should "The Proof", even be in the history section and rename history to something like the birth of Fermat's Last Theorem?

"In 1986, Ken Ribet had proved Serre's epsilon conjecture that every counterexample to Fermat's Last Theorem would yield an elliptic curve defined as:

y2 = x(x - an)(x + bn) which would provide a counterexample to the Taniyama-Shimura conjecture. This latter conjecture proposed a deep connection, now proven to be true, between elliptic curves and modular forms. In one formulation it states that every elliptic curve can be parametrized by a rational map with integer coefficients using the classical modular curve; elliptic curves are all modular." Timothy Clemans 18:17, 27 June 2006 (UTC)

Timothy, you may have noticed that I made some small formatting changes to your edits. Unfortunately, I cannot comment on the technical details. Feel free to put more technical stuff in; in particular, if algebraic number theory came about as a consequence of studying the FLT, that's a good thing to mention. However, bear in mind that we try to keep the lead section short and preferably accessible to non-specialists. You're welcome to go deep in the body of the article. If you prefer, you can rename the section "The proof" to "The connection between elliptic curves and modular forms", or something like that. -- Jitse Niesen (talk) 12:22, 28 June 2006 (UTC)
Much of the research today can be traced to Fermat if not Fermat's Last Theorem(based on a statement by Andrew Wiles printed the Fermat's Engima(USA)) and new mathematics was created to try to prove FLT. Timothy Clemans 22:42, 28 June 2006 (UTC)
Well, that's the dumbest description of FLT I've seen in a while. 23:33, 5 August 2006 (UTC)

Summary for the confused?

Could someone please summarize what's happening on this talk page for me? Someone says Wiles's proof is wrong, but doesn't have a counterexample? It's very hard to follow what with people adding comments in willy-nilly and not indenting properly. I'd prefer a neutral summary, of course, but I should be able to figure out what's going on from a one-sided summary. —Keenan Pepper 02:53, 28 June 2006 (UTC)

The counterexamples are just above your eyebrows on the screen. E. E. Escultura
By "counterexample" I mean positive integers a, b, c, and n > 2 such that an + bn = cn. If you don't have the integers you don't have a counterexample. —Keenan Pepper 22:56, 29 June 2006 (UTC)
The integers have to be fixed first with valid axiomatization before FLT would make sense as a problem on the integers. E. E. Escultura
I have no idea what that means. It's always been a problem on the integers. Are you taking issue with integer arithmetic now?? —Keenan Pepper 05:56, 30 June 2006 (UTC)

The integers as subspace of the real number system are not well-defined because the real number system and its concepts are also not well-defined by consistent axioms. It has no separate adequate axiomatization either. This is the content of my posts here including the new topic below on Contradiction-Free Mathematical Space. E. E. Escultura

Fermat's Last Theorem doesn't say anything about real numbers, or even the integers "as a subspace of the real number system", just integers, by themselves. It's obvious that you don't actually have a counterexample and you're just throwing words around. —Keenan Pepper 07:14, 30 June 2006 (UTC)

To make sense of FLT one needs to well-define the integers by consistent axioms and prove it using the axioms. No one has done that. I well-defined the integers by embedding them isomorphically in the new real number system well-defined by consistent axioms. This makes sense of both FLT, the integers and the new integers. Then the countable counterexamples to FLT are constructed from the integers and new integers. E. E. Escultura

Please indent your replies properly and sign them with a date so we can follow the conversation. First of all, what's wrong with the Peano axioms? Second of all, what the hell does "embedding them isomorphically" mean? I'm familiar with both those words but they don't make sense in this combination. I searched for it but the only results I found were your postings on math forums. —Keenan Pepper 14:12, 3 July 2006 (UTC)

Try proving FLT using Peano's postulates alone. Even Wiles used the real number system but his shortcoming was he did not fix it first. Embedding and isomorhphism are standard techniques in topology and algebra. Either of them subjects the embedded space to the axioms of the embedding space and, therefore, provides axiomatization of the former. These results are new and published; therefore you can't find them in books and older issues of any journal. E. E. Escultura 4 July 2005.

Proving it from Peano's postulates would be one thing, but you claim to have found a counterexample, which you have yet to share with us. A counterexample could be verified using nothing but integer arithmetic. Do you disagree with that statement? BTW, it's considered rude to change your comments after someone else has replied to them, because it can make their reply seem inappropriate. —Keenan Pepper 18:36, 4 July 2006 (UTC)
You have repeatedly mentioned the integers and I have repeadly responded that they integers do not have valid axiomatization at this time. Peano's postulats are inadequate as they are essentially definitions. You need consistent axioms to build a valid (meaning, contradiction-free) mathematical space. Therefore, FLT is nonsense as a problem on the integers and can neither be proved nor disproved. The only way to make sense of FLT is to build the contradiction-free new real number system, which I did using using the three axioms above, and puting the integers (isomorphic embedding). The new real number system now include the integers (which are now well-defined by its three axioms) plus the new integers which are numbers of the form N.99..., N = 0, 1, ... Then reformulate FLT as a problem on the new real number system and you have the counterexamples as stated above.
There is are no known counter-examples to "If n is greater than two, then an + bn = cn has no solutions in non-zero integers a, b, and c." as stated and that is the definition of Fermat's Last Theorem. The theorem has been proved and if one removes the integers from its definition they are no longer dealing with Fermat's Last Theorem. If there counterexamples to FLT formulated in your real number system then FLT in that system is not a theorem. I do not like the notion of a theorem being true and false at the same time. We can not go around saying that your real number system is brand new nor that counter-examples of FLT in other settings are new. FLT from the viewpoint of integers has been proven. I know what an integer is and it is well-defined.
Unless you are talking about the FLT article itself not FLT, then you should not be posting here. We can not put your comments into the article due to the policies of Wikipedia and as a user here you are required to allow the policies like everyone else. It costs money to support this public service that is for developing an open encyclopedia. It is not a news group.
It is one thing to have an idea. It is a completely different thing to have an idea that is well-developed and is supported by popular viewpoint. Timothy Clemans 18:28, 5 July 2006 (UTC)
The following template is what you need to follow:
Also since your problem with Wiles and FLT is the real number system that is used right now, you shouldn't even be here talking it since it is more approiate(sp) there than here. Very few people think that a mathematician should have to rewrite the foundations just to prove a problem about the integers that are the most appected(sp) foundations out there.
Timothy Clemans 18:41, 5 July 2006 (UTC)
Ok, here the Wikipedia view on debates in article talk pages!

What talk pages may be used for Talk pages are not for general chatter; please keep discussions on talk pages on the topic of how to improve the associated article.

Talk pages are also not strictly a forum to argue different points of view about controversial issues. They are a forum to discuss how different points of view should be included in the article so that the end result is neutral. Partisan debates do not align with the mission of Wikipedia, and get in the way of the job of writing an encyclopedia. (For an alternative forum, see the m:Wikibate proposal.) Arguing as a means of improving an article is considerably less effective than an equal amount of time engaged in research.

For issues which have a verifiably correct and relatively undisputed answer, please do feel free to use the talk pages to facilitate fact checking (which sometimes includes resolving disputes over factual accuracy). Timothy Clemans 18:49, 5 July 2006 (UTC)

As I understand it, there is just one person around here who claims that the proof by Wiles is not valid, and he has given some reasons that I find extremely difficult to follow. I think it is fair to say that almost everyone else (both here and in the wider mathematical community) agrees that Wiles is correct. Madmath789 11:08, 28 June 2006 (UTC)

Anyone may believe in anything including the Flat-Earth Theory but science and mathemtics are not matters of belief and agreement; they have high standards and requirements for establishing mathematical and scientific truths. E. E. Escultura

Agree with Madmath. Dr Escultura is using this page to advertise his/her views on the fundamental nature of mathematics. He/she claims not only that Wiles's proof is wrong, but that also in fact a hefty slice of contemporary mathematics is wrong. I believe that his/her views are not widely held. I plan to continue not taking Dr Escultura's comments seriously. Also I predict that Dr Escultura will reply to these comments with several lengthy paragraphs.... I will drink once for each use of the terms "ill-defined", "contradiction-free", "mathematical space". Dmharvey 11:39, 28 June 2006 (UTC)
Like science, its language, which is mathematics, is not built on popular consensus. The arbiter of scientific knowledge, including mathematics and science, is the network of refereed international scientific journals. That is the way it is in mathematics and science. My work is published in over two-dozen scientific papers by over half a dozen renowned refereed international scientific journals. Internet discussions do not count at all as far as scientific knowledge is concerned. The counterexamples to FLT are just above Dmharvey's grey matter on the screen. In mathematics what is ill-defined does not exist, ambiguous, nonsense, etc. Someone here does not know basic mathematics. There is no need to advertise my views; they are published and etched in the archives and databases.
Ideas need to be exposed to amateurs repeatedly before they sink in. That is what makes messages lengthy. At the same time, when an amateur becomes a bit inebriated the chance of sinking in becomes also a bit remote.

E. E. Escultura

**hiccup** Dmharvey 11:57, 29 June 2006 (UTC)
Internet discussions do not count at all as far as scientific knowledge is concerned. Then what are you doing here? Dmharvey 18:48, 28 June 2006 (UTC)
Scientists and mathematicians share their contributions to help elevate the human condition. It is also an opportunity for the public and colleagues to learn, scrutinize and make a critique of their contributions. If they want their critique to become part of scientific knowledge they must publish their analysis and findings in refereed journals. Otherwise, they may not even make it to the footnote of scientific history. E. E. Escultura
Dude it's very simple. Wikipedia is "built on popular consensus". It's designed that way from the ground up. Indeed, if you attempt to elevate the human condition by adding your "analysis and findings" to the actual article (rather than just the talk page), I'm sure you will find (as you have found before) that it will get removed. Clearly, what you're trying to do by arguing your case on the talk page is bring the wikipedia community around to your point of view, so that your material can be safely added to the article itself. Judging by the material you have produced so far, I promise that there are quite a few wikipedians of a mathematical bent (including myself) who will not be swayed. Your refusal to get an account on wikipedia certainly doesn't help your credibility in these circles.

Folks may believe anything but mathematical and scientific issues are not resolved by popular consensus. There are high standards for resolving them. E. E. Escultura

Since last year several blogs have attacked my work on the wrong issue. Unfortunately, I noticed them only just recently. I have started responding to them, explained the issue and put my views forward. That's how I got here. Unfortunately some blogs cried uncle. I won't stay longer here than necessary. E. E. Escultura

I've looked you up on MathSciNet, and I can report the following. You do indeed exist (or at least there is a guy called E. E. Escultura who has published a bunch of papers, which we can presume is you). There are ten papers listed. There are two papers with "Fermat" in the title: "Exact solution of Fermat's equation (a definitive resolution of Fermat's last theorem)" in 1998, and "Probabilistic mathematics and applications to dynamic systems including Fermat's last theorem" in 1996. There is only one citation listed for the first article, and it's from a paper you wrote in 2000. There are no citations for the second article. Nobody has even bothered to review these papers for the AMS Mathematical Reviews. (Geez, I've only ever had one little crappy paper published, where I didn't try to do anything particularly original, and they managed to find someone to review that.) One would suspect that if your ideas are as important and revolutionary as you make them out to be, then in the last ten years there would be more coverage of them in the "network of refereed international scientific journals".

It takes time to learn and grasp new developments and even longer to review and use them in research. I assure you that the experts are studying my work. One of my papers last year got into No. 8 among the top 25 most downloaded papers of the 2,000 titles published by Elsevier Science last year. Visit this link: Morever, I have been invited lately to keynote international conferences. The abstract of one of my keynote addresses is online at the Atlas website: E. E. Escultura

Anyway, I've gotten into stupid discussions like this in the past, and I've ended up wasting a ton of my time... so I'm going to take a very deep breath now, and promise to you and myself that I will not respond further to any more comments you make here. I will of course revert any inappropriate changes you make to the actual article. Good luck with that whole human condition elevation thing. Dmharvey 11:57, 29 June 2006 (UTC)

I won't stay here longer than necessary. E. E. Escultura

One of the reasons why hate this axiom stuff of real numbers is that I do not like arguments from the social world. Besides, there was a guy who told me that Andrew Wiles was dumb, because he did not FLT in the normal version, but the version in line with the abc conjecture. Then there is my math teacher who seems to think that induction and mathematical induction are the same and are both bad to use in mathematics. One induction is used to form conjecture. Mathematical induction is used to prove statements about the natural numbers from a base with an argument as to why the induction step is vaild in that stitution. I just want to get to a university, but hey geometry at community college is fun and my teacher now knows that the CRC handbook is online at and that "How to Solve It" exists and is the best mathematics book for the student and teacher I have ever read. I want to hear from more normal people that would be in the same class as say Ken Ribet. BTW, the think EE is about 50 years old. Anyways, there is all kinds of trash related to his stuff agaisnt the proof of Fermat's Last Theorem or "I am sueing Andrew Wiles". I love the world wide web, because I access to this stuff. Timothy Clemans 20:40, 29 June 2006 (UTC)

Mathematical axioms have nothing to do with social issues unless they are designed to serve social science. They are the basis of mathematical proofs, arguments and analysis. E. E. Escultura

The following seems incorrect: "If they want their critique to become part of scientific knowledge they must publish their analysis and findings in refereed journals." That's strange, because we know about flaws in proofs that others found, but they did not published their findings related to those flaws such as in the case of the Euler argument in the first manuscript of "Modular elliptic curves and Fermat's Last Theorem". Timothy Clemans 20:46, 29 June 2006 (UTC)

Results have to be published in scientific literature to be recorded in scientific history. E. E. Escultura

Does mathematics even follow the rules of scientific documentation? Mathematics is not science, because it is not based on the scientific method. Timothy Clemans 20:46, 29 June 2006 (UTC)

Yes, mathematics is a language whose grammer consists of the rules of inference. Like science it has high standards of documentation and verification. E. E. Escultura

Does Dr. Escultura know User:BenCawaling? I have a feeling they'd really hit it off. —Keenan Pepper 14:49, 28 June 2006 (UTC)
 :-) Dmharvey 14:52, 28 June 2006 (UTC)

Prescription for Contradiction-Free Mathematical Space

The following subjects are offshoots of the critique-rectification of the underlying fields of FLT, namely, foundations, number theory and the real number system:

1) Contradiction reduces a mathematical space to nonsense; among the sources of contradiction are the following: ambiguity or uncertainty, ill-defined concepts, vacuous statement, large or small number depending on context (due mainly to limitation of computation) and self reference, e.g., the indirect proof. Therefore, I enumerate below some of the requirements for avoiding contradiction.

2) The first major step in avoiding contradiction was done by Hilbert a century ago when he recognized that the concepts of individual thought cannot be the subject matter or objects of a mathematical space since they are inaccessible to others and can neither be studied or discussed collectively nor axiomatized.

3) Therefore the mathematical objects must be symbols in the objective world (we call them concepts, too) well-defined by consistent axioms. Now, I require that every concept must be well-defined, i.e., its existence, properties and relationship with other concepts be specified by the axioms. Existence is stressed since vacuous statement or defining expression is inherently contradictory, e.g., the statement: “the solution of the equation x^2 + 1 = 0 is not 0” is quite true but it leads to the contradiction 1 = 0; the reason for it is: the equation is vacuous. That solution has been denoted by i = sqrt(-1) = sqrt(1/-1) = 1/i = -i, from which follows that 2i = 0 or 1 = 0.

4) Since every mathematical space is well-defined only be its own axioms distinct mathematical spaces are independent; therefore, a concept in one is ill-defined in the other and any proposition involving concepts from two distinct spaces is ambiguous or ill-defined.

5) It follows from 4) that the rules of inference must be specific to and well-defined by the axioms of the given mathematical space. In particular, formal logic is flawed because it applies to distinct mathematical spaces.

6) It also follows from 4) that undecidable propositions are those involving ill-defined concepts. In particular, Goedel’s incompleteness theorems are nonsense because they involve concepts from two distinct spaces, the integers and the propositional calculus.

7) A decimal is well-defined only if every digit is known or computable, i.e., there is some rule or algorithm for determining it uniquely. The numbers pi and the logarithmic base e are well-defined since their digits can be computed from their series expansion. So is a normal number whose digits are chosen at random from the basic integers.

8) A mathematical space must be built on finite concepts to avoid the ambiguity of infinite set. Thus, I built the new real number system (the decimals) on the basic integers, 1, 2, …, 9. Then I well-defined the decimals, terminating or nonterminating.

9) Ambiguity may be introduced provided it is “approximable” by certainty; for example, a nonterminating decimal is ambiguous (except in exceptional cases) because not all its digits are known but it can be approximated to any desired margin of error; for example a nonterminating decimal is approximated by its initial segment at the nth decimal digit and the margin of error is 10^-n.

10) Here is an example of self-reference called the Richard paradox: The barber of Seville shaves those and only those who do not shave himself. Who shaves the barber? E. E. Escultura

LOL, I love how #10 is just tacked on the end there, and it's not Richard's paradox, it's the barber paradox, attributed to Bertrand Russell. Also, I love how a "number whose digits are chosen at random" can be "well-defined", that's a hoot. —Keenan Pepper 07:21, 30 June 2006 (UTC)
A decimal is well-defined if every digit is known or computable, i.e., there is some rule or algorithm or scheme for determining each digit uniquely. In a normal number every digit is chosen from the basic integers at random. Thus, the choice of each digit is unique. E. E. Escultura
The barber paradox is the Richard paradox; The Russell paradox refers to a Cretan saying, All Cretans are liars. Is he telling the truth? Anyway the point here is: they are both self-referent. Russell did not know this; that's why his remedy did not work.
Out of (fairly mild) curiosity: does anyone know where I can actually get to see one of the many (refereed) papers that Escultura has written? I would quite like to read some of his ideas when properly published, but I have been unable to find any of his papers available online (at least without cost, and I am not that keen!) Madmath789 07:53, 30 June 2006 (UTC)
Use your browser to visit the websites of the journals listed here: Nonlinear Analysis, Problems of Nonlinear Analysis in Engineerings Systems, Applied Mathematics and Computation, Indian Journals of Pure and Applied Mathematics, Nonlinear Studies, International Journal of Nonlinear Differential Equations, Nonlinear Analysis and Phenomena, etc. You will be instructed on how to access the papers. E. E. Escultura
Been there and done that, but I can only find papers that you have to pay for. If you can provide me with a URL to a paper which is accessible online, I would be grateful. Madmath789 10:38, 30 June 2006 (UTC)

I'm sorry but the publishers of the journals sell their articles. Elsevier Science sells theirs for $30/article. E. E. Escultura

No problem - I now have some (free) copies from the Indian Journal of Pure And Applied Mathematics ( - and very interesting reading they make, too! Madmath789 12:08, 30 June 2006 (UTC)

Historical note ?

Maybe it is appropriate to add this little historical note on this article: Fermat was a French lawyer who dabbled in mathematics as a hobby through correspondence with friends. I'm not aware of any mathematical paper he published but he is supposed to have wriiten a paper on optics. In 1637 he wrote a note on the margin of Diiophantus' book, Arithmetica (3rd Century, AD, I think), to the effect that he discovered a truly remarkable theorem but the margin was too narrow for the proof. What is now known as Fermat's equation stated above is one of the Diophantine equations. It's really an indeterminate equation in three unknowns. E. E. Escultura, July 8, 2006(date is incorrect more correct as July 6 or earlier Timothy Clemans 17:26, 6 July 2006 (UTC) please use wiki code to put date in)

No need for this note. More extensive background on Fermat is given in the Pierre de Fermat article, where it belongs. Considering the many areas of mathematics that he contributed to, your description of him as a "lawyer who dabbled in mathematics" is misleading. And you should learn how create a new sub-head when you start a new topic (as well as indenting your posts, talk page etiquette etc.). Gandalf61 08:30, 6 July 2006 (UTC)

Diophantine equations and FLT

He did not write formal papers but did extensive correspondences with friends; nor did he have formal training in mathematics. He was a lawyer by profession. This is not to downplay his great contributions to mathematics. 21:18, 6 July 2006, E. E. Escultura

I'd like to add that he had extensive mathematical notes in his private files. 0:48, 8 July 2006, E. E. Escultura

Fermat did not do his work through others. xn + yn = zn is not a Diophantine equation however x2 + y2 = z2 is. Much of number theory today goes from Fermat. See Fermat's little theorem. I don't want to see a misleading article, because anyone could say well Fermat's Last Theorem must be wrong, because some hobbiest stated it. Timothy Clemans 17:24, 6 July 2006 (UTC)
Not sure what is meant by "Fermat did not do his work through others", but it is wrong to say that xn + yn = zn is not a Diophantine equation - it most certainly is! Madmath789 18:05, 6 July 2006 (UTC)
Mathematicians Henri Darmon, Fred Diamond, and Richard Taylor(Elliptic Curves, Modular Forms & Fermat's Last Theorem edited by J. Coates and S.-T. Yau) state that x, y, and z are integers but n is not labeled an integer by them. The equation xn + yn = zn is only a Diophantine equation if we say that its solutions are required to be integers.

"What is now known as Fermat's equation stated above is one of the Diophantine equations. It's really an indeterminate equation in three unknowns." - what about n(Is it not an unknown too?)

No, n is a given integer greater than 2. 21:58, 7 July 2006, E. E. Escultura

What we care about is "If n is greater than two, then an + bn = cn has no solutions in non-zero integers a, b, and c." We could say that there are no solutions to a2.5 + b2.5 = c2.5 in non-zero integers a, b, and c is a Diophantine equation but not for the Fermat equation. Timothy Clemans 18:31, 6 July 2006 (UTC)

The statement makes sense only if the integers are correctly axiomatized as a mathematical space so that proof or disproof based on the axioms would be valid. 9:03, 8 July 2006, E. E. Escultura
I have now corrected the erroneous dates of some of my posts and checked the indentation, too. 0:41, 8 July 2006 E. E. Escultura

Dead mathematicians ?

I strongly remember reading in a book titled Fermat's Last Theorem many years ago that over the years half a dozen or more mathematicians have commited suicide over the problem. I do not know wether this is true (I always assumed it was), but if any of you pros know more about it I think it should be added. J. Stoever 15:26, 23 July 2006 (UTC)

I think I've heard that. If it is true then I think it should be added, because work on FLT has had hudge affect on various people and society(unsure). Timothy Clemans 22:25, 23 July 2006 (UTC)

Fermat's proof stuff is annoying me

Why not just do a section on elementary proofs instead of did the guy who inspired all the work on FLT write an elementary proof that was valid and yet many people just have not rediscovered such a proof. There are more important things to be working on in this article like fixing the references section and math formulas. Timothy Clemans 07:14, 24 July 2006 (UTC)

simple and elementary proof of fermats last theorem is possible.

like , like , , here when "a" is any natural number "b" is any natural number other then the same value of "a" and "n" is any prime then "k" will always be a natural number. from the above information we can prove very easily and in an elementary way that it is impossible to write

                                               for n>2.Bhowmickr 14:32, 26 August 2006 (UTC)
If you would like to explain your proof here, I am sure there are many people here who will explain why it fails. Madmath789 13:22, 26 August 2006 (UTC)
what if it doesnot fails.Bhowmickr 14:32, 26 August 2006 (UTC)
Please, let's not respond to this troll. Ryan Reich 14:05, 26 August 2006 (UTC)
Actually I'm mildly interested. Bhowmickr, please supply the missing portion of your proof so we can discuss it. We can take it to my talk page if people object that we're not talking about how to improve the article. —Keenan Pepper 02:05, 30 August 2006 (UTC)


I am starting to doubt that you really ar Escultura, because your internet is out of Australia. You need a user account so that there is one place to write to you for Wikipedia perpuroses. On Wikipedia, I only care about popular view and view of the integers is not popular. You need to stop talking about the integers on this page. Timothy Clemans 17:48, 7 July 2006 (UTC)

I couldn't care less about anyone's opinion. Im posting as a mathematician and I'll respond to any attack on my work and comment on mathematical points. I travel a lot and part of my family is in Australia; if you want my e-mail, here it is: 9:48, 8 July 2006. E. E. Escultura
You do not own Wikipedia therefore you must follow the rules here. Why did you change your email from domain name yahoo to hotmail? Why do you not have a Wikipedia user account? With the way Wikipedia works, you need to use wikicode to sign your name and put in the correct date and time using the UTC standard and Wikipedia's clock. Timothy Clemans 18:48, 8 July 2006 (UTC)
University of the Philippines Pampanga, Clark Field, Pampanga, Philippines; Residence: Blk 1 Lot 1 Granwood Villas, BF Homes, Q. C. 1120, Philippines;
Timothy Clemans 18:51, 8 July 2006 (UTC)
I am just a blogger and I don't know the technical matter of creating an account in Wikipedia. I want my Yahoo account to be exclusively for scientific correspondences so that it won't be clogged with junk mails. Anyway, if there is nothing here that concerns my work I won't post anything. 9:00, 10 July 2006. E. E. Escultura
Escultura, it's very easy to create an account. Just click on "create account" in the top-right corner and follow the instructions. Dmharvey 00:59, 10 July 2006 (UTC)
If you were able to create accounts for Yahoo and Hotmail, then why do you think that it would be hard to create an account for Wikipedia? "Anyway, if there is nothing here that concerns my work I won't post anything." and yet you came here and started the "Wiles proof is wrong" thing(at least I think you did(I could check)). It is a really bad idea to use a Wikipedia article talk page as a blog! Timothy Clemans 18:46, 10 July 2006 (UTC)

I have an account already and I'm logging in from it. You are wrong. This talk page started calling me names last year but some of them were deleted, according to a blog, before I noticed some remaining ones a couple of months ago. That was when I posted the message on Wiles' wrong proof. You were the one calling me names and I never did. I just know that those with empty top resort to name-calling and pull out ideas from the flat of their foot. E. E. Escultura

I would like to finally prove Escultura wrong, in his claim that Wiles did not prove "Fermat's Last Theorem" true. There is a book, authored by the reknowned physicist Roger Penrose, titled, "The Road to Reality," that in the third section of chapter one (denoted as 1.3 in the upper margin), Penrose clearly states that Wiles did prove "Fermat's Last Theorem" on his second attempt. Unless Escultura, believes he has greater knowledge and insight than Roger Penrose (a man who has made great many contributions to science and mathematics).

Penrose and his renown have no bearing on anything. Rather, Escultura was proved wrong by Andrew Wiles providing a proof of the theorem. There is no obligation on anyone to prove Escultura wrong; rather, the burden is on Escultura to prove his assertion. -- Jibal 05:55, 29 December 2006 (UTC)
I did not assert anything. I PUBLISHED the counterexamples to FLT to prove the conjecture wrong. Unless a PUBLISHED REFUTATION comes along the counterexamples stand.

E. E. Escultura

A true mathematician does not believe or disbelieve what someone says no matter how knowledgeable but examines what is said to find out if it makes sense. —The preceding unsigned comment was added by (talkcontribs) 2006-11-16T19:39:04.

Please split talk page up

This talk page is now 91KB in size. Please could someone who knows how archive all inactive discussions. Tompw 19:09, 7 October 2006 (UTC)

Done. There are instructions at WP:ARCHIVE. CMummert 00:55, 17 November 2006 (UTC)

Can Wiles' proof be carried out in second order arithmetic?

I am unaware of any published literature which shows that Wiles' proof can be carried out in second-order arithmetic (and thus no literature saying the proof can be carried out in PA). The best reference I have seen is an FOM discussion that was formerly referenced from the article. Please post references to any such published literature below this message. If no reliable sources can be found in a reasonably long period of time, the article will have to be modified. CMummert 00:24, 20 November 2006 (UTC)

Since no citation is forthcoming (and I have no idea where to find one), I have moved the following from the article. The prose was added by User:R.e.b, who suggested that removing it is OK.
These constructions use axioms that go beyond Zermelo-Frankel set theory (ZFC), which has led to a myth that Wiles' proof is not carried out in ZFC. In fact Wiles only uses a tiny piece of this machinery, only involving étale cohomology of schemes of finite type over prime fields, and this can be carried out in second order arithmetic, a much weaker theory than standard set theories. Everything in his proof can be done in second order arithmetic (and could probably be encoded in first order (Peano) arithmetic, though this would require considerable effort).
The issue of whether the proof can be formalized in ZFC or SOA is interesting, but not of central importance, since there is widespread acceptance of the proof. In order to keep the article verifiable, and since there is some disagreement in the professional community about the issue, I hope other editors will refrain from adding additional material to the article about the formalization of the proof unless a published reference for the material can be provided. CMummert 17:57, 7 December 2006 (UTC)

Wiles proof error?

However, no correct proof was found for 357 years, until it was finally proven using very deep methods by Andrew Wiles in 1995 (after a failed attempt a year before).

Wasn't the original proof proposed in 1993, 2 years before? 17:38, 20 November 2006 (UTC) Matty_B

This is just a minor editing of the above statement. A proof is no proof until it has overcome all challenges and criticisms and Andrew Wiles' proof has plenty of them both in the scientific literature and in the many blogs, fora, threads, chatrooms and websites across the internet that have not been refuted. Until that time it remains, at best, controversial. 12:54, 19 December 2006 (UTC) E. E. Escultura

No, a proof is a proof when it is valid. And there is no burden to refute the unending stream of "challenges and criticisms" on blogs, etc.; it is the challenger's burden to prove their assertions, which they cannot do because FLT, controversial or not, has been proved. -- Jibal 05:50, 29 December 2006 (UTC)
A PUBLISHED challenge to a proof requires PUBLISHED refutation to shoot down. Otherwise, the challenge stands. Who passes judgment on the validity of a proof? Furthermore, the published counterexamples to FLT showing that the conjecture is false require published refutation to shoot down. Otherwise, they stand. E. E. Escultura

Fermat's original proof

People worrying about what happened to the original proof by Fermat and what it could have contained can take comfort in that the proof might never have existed. Fermat could have been lying, or his proof could have been erroneous. JIP | Talk 12:00, 23 November 2006 (UTC)

I'm not clear on how this would give anyone comfort, or why you think that anyone might not have considered this before you suggested it. -- Jibal 06:03, 29 December 2006 (UTC)

Proposed correction

In section "Proof" the end of the first paragraph states: "This theorem said that every counterexample an + bn = cn to Fermat's Last Theorem would yield an elliptic curve defined as y2 = x(x − an)(x + bn) which would not be modular and therefore provide a counterexample to the Taniyama–Shimura conjecture. Fermat's Last Theorem and Taniyama-Shimura were now linked through the Epsilon conjecture; either both were true or both were false."

I believe the last sentence is not true. It should be: "Fermat's Last Theorem and Taniyama-Shimura were now linked through the Epsilon conjecture; the (conjectured) truth of Taniyama-Shimura was shown to imply the truth of Fermat's Last Theorem."

The difference is that even if Fermat's Last Theorem was shown to be true by some other means, it would only eliminate this particular counterexample to Taniyama-Shimura, which might have other counterexamples. Its truth would not be immediately implied by the truth of Fermat's Last Theorem.

You are correct there. The fact that the falsity of FLT implies the falsity of T-S is simply the contrapositive of the result. However, I don't much like your phrasing, for the reason that "conjecture" appears a little too often. It makes the statement seem tentative. What I would suggest is: "Fermat's Last Theorem and Taniyama-Shimura were now linked through the proof of the Epsilon conjecture; the truth of Taniyama-Shimura was shown to imply the truth of Fermat's Last Theorem." The fact that Taniyama-Shimura was conjectured to be truth is not really relevant to the implication, so I would definitely remove the parenthetical remark; and adding "the proof" before "Epsilon conjecture" would remind readers that, despite the name, it was then an established fact. At least, that's my suggestion. Magidin 15:07, 1 December 2006 (UTC)
I agree with your phrasing. I was only trying to draw attention to the fact that at this point in time the truth of T-S had not yet been settled, but of course that is not essential to the argument, so your proposal improves on it. Please go ahead and make this change as soon as it is convenient, as I do not have a wikipedia account, and I am not sure how long we must wait to give others a chance to comment on it. Thanks.
Isn't creating an account just a matter of choosing a user name and password and entering them into the appropriate boxes? But you don't need an account to edit the article, and you don't have to wait for anyone - you could have made the change when you first spotted the error (WP:Be bold). --Zundark 17:06, 1 December 2006 (UTC)
I was the one who wrote that line; it was meant to clarify things for those non so adept in the higher levels of math. However, you are correct; you're phrasing is superior.--HereToHelp 01:58, 20 December 2006 (UTC)